As proposed by Hosoya, the Hosoya index (also called Z-index) of a graph is defined by Z | = | sum_(k = 0)^n left bracketing bar a_k right bracketing bar | = | sum_(k = 0)^n b_k, where n is the number of vertices of the graph, a_k is the kth coefficient of the matching polynomial, b_k is the kth coefficient of the matching-generating polynomial, and left bracketing bar x right bracketing bar is the absolute value of x. In others words, it is just the number of independent edge sets (i.e., matchings) in a graph.
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